Eliminating Illusion in Directed Networks

TL;DR

This paper investigates the computational complexity of eliminating majority illusion in directed social networks through vertex recoloring, revealing NP-hardness in general but polynomial solutions in specific structured networks.

Contribution

It establishes NP-hardness results for the illusion elimination problem and provides polynomial algorithms for certain sparse and structured network classes.

Findings

NP-hardness for p=1/2 even on grid networks

NP-hard and W[2]-hardness when parameterized by recoloring count

Polynomial algorithms for outerplanar networks, trees, and cycles

Abstract

We study illusion elimination problems on directed social networks where each vertex is colored either red or blue. A vertex is under \textit{majority illusion} if it has more red out-neighbors than blue out-neighbors when there are more blue vertices than red ones in the network. In a more general phenomenon of $p$-illusion, at least $p$ fraction of the out-neighbors (as opposed to $1/2$ for majority) of a vertex is red. In the directed illusion elimination problem, we recolor minimum number of vertices so that no vertex is under $p$-illusion, for $p\in (0,1)$. Unfortunately, the problem is NP-hard for $p =1/2$ even when the network is a grid. Moreover, the problem is NP-hard and W[2]-hard when parameterized by the number of recolorings for each $p \in (0,1)$ even on bipartite DAGs. Thus, we can neither get a polynomial time algorithm on DAGs, unless P=NP, nor we can get a FPT algorithm even by combining solution size and directed graph parameters that measure distance from acyclicity, unless FPT=W[2]. We show that the problem can be solved in polynomial time in structured, sparse networks such as outerplanar networks, outward grids, trees, and cycles. Finally, we show tractable algorithms parameterized by treewidth of the underlying undirected graph, and by the number of vertices under illusion.